Besides the above two conserved laws of the Euler equations (REF ), one can find others in [1]}.
In a seminal paper [2]},
Onsager conjectured that weak solutions of the Euler equation with
Hölder continuity exponent \(\alpha >\frac{1}{3}\)  do conserve energy and that turbulent
or anomalous dissipation occurs when \(\alpha \le \frac{1}{3}\) . The first attempt to the energy conservation issue for weak solutions of the Euler equations (REF ) was given by Eyink [3]} in a stronger Hölder space \(C_*^\alpha \)  with \(\alpha >\frac{1}{3}\) ,
which is included in \(C^\alpha \) . Subsequently Constantin-E-Titi [4]} successfully solved the positive part of the
Onsager's conjecture and proved that the energy is conserved if a weak solution \(v\)  is in the Besov space \(L^{3}(0,T; B^{\alpha }_{3,\infty }(\mathbb {T}^{3}))\)  with \(\alpha >1/3\) . Later on, by deriving a local energy equation which contains a term \(D(v)\)  representing the dissipation or production of energy caused by the lack of smoothness of solution, Duchon and Robert gave a weaker condition on the solutions to conserve energy in [5]}, that is, if \(v\)  satisfies \(\int |v(t,x+\xi )-v(t,x)|^3dx \le C(t)|\xi |\sigma (|\xi |)\) , where \(C(t)\in L^1(0,T)\)  and \(\sigma (a)\rightarrow 0\)  as \(a\rightarrow 0\) , then energy is conserved. Recently,
Cheskidov-Constantin-Friedlander-Shvydkoy [6]} refined the above spaces to the
critical space \(L^{3}(0,T; B^{1/3}_{3,c(\mathbb {N})})\) , where
\(B^{1/3}_{3,c(\mathbb {N})}=\lbrace v\in B^{1/3}_{3,\infty }: \lim _{q\rightarrow \infty }2^{q}\Vert \Delta _{q}v\Vert ^{3}_{L^{3}}=0\rbrace \)  and \(\Delta _{q}\)  represents a smooth restriction of \(v\)  into Fourier modes of order \(2^q\) . In the follow-up paper, Shvydkoy [7]}, [8]} stated that this condition of energy conservation can be equivalent to
\(\lim _{z\rightarrow 0}{|z|}=0.\) 
